All Questions
Tagged with polygons euclidean-geometry
158
questions
3
votes
2
answers
355
views
Collinearity in bicentric pentagon
Can you provide a proof for the following claim:
Claim. The circumcenter, the incenter, and the excenter of the pentagon formed by diagonals in a bicentric pentagon are collinear.
GeoGebra applet ...
- 12.9k
1
vote
1
answer
905
views
Constructing an isosceles trapezoid with a specific decomposition into triangles
A recent question asked about finding the ratio of the bases for the following isosceles trapezoid:
That problem has been solved, obtaining a result of $|CD|/|AB|=1-1/\sqrt{2}$. What I'm curious how ...
- 16.8k
0
votes
2
answers
117
views
Excircle and parallelogram
Can you provide a proof for the following claim:
Claim. Given any parallelogram $ABCD$ and excircle of triangle $\triangle ABC$ oposite to vertex $A$. An arbitrary tangent $t$ is constructed to the ...
- 12.9k
2
votes
1
answer
167
views
An ellipse determined by circumcenters
Can you prove or disprove the following claim:
Claim. A convex hexagon $ABCDEF$ is circumscribed about an ellipse. Let $G$ be the point of concurrency of hexagon's principal diagonals , and let the ...
- 12.9k
-2
votes
1
answer
58
views
How to calculate the number of side of a polygon?
I stuck on this problem. Please suggest any hint on how to solve this problem. Thanks in advance.
- 379
5
votes
1
answer
510
views
Simple proof of "maximum number of right angles in a convex $n$-gon is 3 for $n\geq 5$" for a 8th grade student?
I know a proof of "maximum number of right angles in a convex $n$-polygon is 3 for $n\geq 5$" as follows:
Suppose $k$ is the number of right angles. Then $180(n-2)-90k$ is the sum of other $...
- 8,571
1
vote
0
answers
19
views
Intersection of Symmetric Convex Sets
Given a symmetric convex compact set $K\in\mathbb{R}^2$, show there are no sequences $(r_i)_{i=1}^n$ of positive scalars and $(P_i)_{i=1}^n$ of points in $\partial K$, so that $n>1$ and
$$\sum_{i=1}...
- 630
1
vote
1
answer
58
views
Characterization of the center of a polygon
Let $O$ be the circumcenter of the regular polygon $P_n$. Then for any $A\in P_n$ one has $d(A,O)\leq r$, where $d$ is the usual Euclidean distance and $r$ is the polygon's circumradius.
Prove that ...
- 342
3
votes
1
answer
370
views
A conic inside a hexagon
Can you prove or disprove the following claim:
Construct a hexagon circumscribed around a conic section. Intersection points of its non-principal diagonals lie on a new conic section.
GeoGebra ...
- 12.9k
2
votes
1
answer
135
views
Formula for the area of a regular convex pentagon
This question is closely related to my previous question.
Can you provide a proof for the following claim:
In any regular convex pentagon $ABCDE$ construct an arbitrary tangent to the incircle of ...
- 12.9k
-2
votes
2
answers
158
views
Existence of regular $n$-gon whose vertices are arbitrarily close to integer coordinates
I'm self-studying these days about polytopes and I came with this question. I don't know if it's true or not.
Let $\alpha_1$, $\ldots$, $\alpha_n$ angles of convex $n$-gon, $n\not=4$. Prove that for ...
0
votes
1
answer
112
views
Vertices of a regular polygon in the plane with irrational coordinates
Today I was trying to solve a problem about triangulation and I came with this
For which $n≥3$ is it possible to draw a regular $n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have ...
3
votes
2
answers
281
views
The ratio of the area of two regular polygons
The polygons in the figure below are all regular polygons(regular heptagon), share a vertex and the orange line crosses the three vertices of the two regular polygons, the area of the small regular ...
- 257
1
vote
0
answers
73
views
How to constrain a rectangle within an arbitrary 2d polgyon?
I am working on a graphics project wherein I need to keep a rectangle (assume it has a fixed but arbitrary size, rectWidth * rectHeight) constrained inside an arbitrary polygon.
The polygon is ...
- 153
1
vote
1
answer
67
views
Circles and enneagon
Using that in a triangle ABC, $\tan\frac A2=\frac{r}{p-a}$ where $p=\frac{a+b+c}{2}$, I found that the radius are equal if
$\tan^220°=\frac{\frac{1+2\cos40°}{\cos20°-1}}{1-\frac{1}{2\cos40°}+\frac{\...
- 513